Global optimization of low-thrust, multiple-flyby trajectories is one of the most challenging problems in interplanetary mission design. The challenge is to develop a method that can design a complex trajectory with little or no human oversight. In the last few years several authors have developed such methods. To date each of those methods required many simplifications of the low-thrust trajectory model in order to perform a global search. In this work a global optimization framework, the Evolutionary Mission Trajectory Generator (EMTG), is presented. A medium-fidelity model based on the well-known Sims-Flanagan transcription is combined with a global optimization heuristic to generate an initial guess for a medium-high fidelity optimization using an integrated trajectory model. When combined, these two methods allow a trajectory designer to rapidly evaluate a wide range of solutions to complex problems.